Question

1. What is the purpose of using a confidence interval? In other words, what does a confidence interval estimate?

2. Why is there a level of confidence associated with a confidence interval? In other words, why isn't a confidence interval 100% accurate?

Answer 1

Part 1

Imagine that you want to know a characteristic of a population, for example, the proportion of people in the country who are in favor of a candidate C. We will call this characteristic "parameter" of the population. For obvious reasons, it is very difficult and expensive to know the opinion of all people in the population.
A possible solution is to take a sample of the population that we consider "representative" and calculate the proportion of people in favor of the candidate in that sample. We will call this value "point estimator of the parameter".
If the sample is a good representation of the population, then the estimator will be very close to the parameter, but we cannot know exactly how good the estimator is because we do not know how good the sample is. To solve this, instead of giving a point gauge, an estimator is calculated per interval and thus be able to give a measure of reliability of being "close" to the population parameter.
In summary, what the interval estimator does is give a confidence of X% of containing the true parameter. This means that: if 100 researchers each take a sample from the population and calculate X% confidence intervals, using the same methods, then the number of intervals containing the population parameter is expected to be X.
NOTE: it is not correct to say that the parameter is or falls within the confidence interval, because the parameter is a fixed value, or that the limits of the interval vary depending on the sample and the confidence....

Part 2

As mentioned before, the confidence intervals depend on how representative the sample taken from the population is.
If the sample is 100% representative, then we would obtain 100% confidence intervals regardless of their amplitude, in fact, it would not be necessary to calculate confidence intervals, with the point estimator it would be enough to know the true population parameter.

The problem is that it is very difficult, or impossible, to know if a sample is an exact representation of the population, it is for this reason that the interval estimation is associated with a confidence level.

This tells us that to obtain 100% confidence intervals we have two options:
1) Study the entire population, which is not always possible, and if it were, it would not make sense to calculate a confidence interval.
2) Give an interval large enough to include all possible theoretical values ​​that the parameter can take, but this would be useless. It is like saying that we are 100% sure that when rolling a die the top face shows a number between 1 and 6, which is obvious.