Confusion regarding Confidence Intervals

Suppose we have $$2$$ independent population parameters $$p_1$$ and $$p_2$$, such that the $$90$$% ( symmetric ) confidence intervals for $$p_1$$ and $$p_2$$ are given by $$(0.411, 0.498)$$ and $$(0.473, 0.567)$$ respectively. Now, according to the original question ( copied verbatim ) , supposing that the $$90$$% interval for $$p_1$$ lies entirely below the interval for $$p_2$$, how confident are we to conclude that $$p_1 < p_2$$?

Now, assuming that I have understood the problem correctly, I feel that this question does not make sense to me because $$(0.411, 0.498)$$ does not lie completely behind $$(0.473,0.567)$$. The question is, am I right in claiming so?

Next, assuming the problem is simply "How confident are we to conclude that $$p_1 < p_2$$ ", given their $$2$$ $$90$$% symmetric confidence intervals as above, how would we go about approaching it, if it is even possible?

Now, I do not think you can talk about "How confident are we to conclude that $$p_1" when no confidence interval is involved: $$p_1 < p_2$$ that will be either true or false. Perhaps from a Bayesian point of view you could talk about the posterior probability that $$p_1 < p_2$$ given that something happens, but that is an entirely different sort of statement than what goes under the name of confidence.
• Yes, so what? My answer was that you cannot properly talk of "the confidence that $p_1 < p_2$". Commented Apr 19, 2020 at 16:20