Here is the exact wording of the question I am struggling to answer.
The manager of a store wants to know more about the proportion of customers who are visiting the store for the first time. She collects a random sample and builds a $90$% confidence interval for the proportion of customers who are visiting the store for the first time: $[0.23,0.29]$.
The regional managers asks what the probability is that the true proportion of customers is higher than $0.3$.
Please choose the most accurate answer.
(a) The probability is exactly $0.05$.
(b) The probability is less than $0.10$.
(c) The probability is greater than $0.10$.
(d) The probability is greater than $0.05$.
(e) The probability is less than $0.05$.
The true proportion $p$ of customers who're visiting the store for the first time is not a random variable, which is the main source of my struggle with this question. This question only makes sense when you interpret it from a Bayesian point of view, but Bayesian stats were not taught in this course.
If we choose to interpret our $90$% confidence interval as saying $\mathbb{P}\left(p\in [0.23,0.29]\right)=0.9$ then the most we can say about the event $p\geq 0.3$ is that it occurs with a probability less than $0.1$.
Am I missing something here? Should we make any assumption about the symmetry of the distribution of $p$?
The answer is (f) None of the above. The reason is as follows: under a non-Bayesian assumption that the true proportion of customers visiting the store for the first time is an unknown constant (which seems reasonable since the manager's interval was called a confidence interval rather than a 'credible interval'), this constant is either above 0.3 or it is not. There is nothing random, so no probability except the trivial '0 or 1'.
Alternatively, if the regional manager wants a Bayesian posterior probability, then any of (a), (b), (c), (d) or (e) could be the most accurate answer, it just depends on the manager's choice of prior.