I'm sharing an example—of an answer to Confidence interval for Uniform($\theta$, $\theta + a$) generated by ChatGPT. I think it will be agreed that it's either wrong or nonsensical in large part & entirely unhelpful; while being to all appearances worth spending one's time in reading.
The issue here is that the confidence interval you derived relies on
the assumption that $Z - Y \le a\sqrt[n]\gamma$.
[It's not an assumption at all, but a statement of when the confidence set is empty.]
If this assumption
does not hold, then the interval $[Z -ad_2, Y -ad_1]$ is not
guaranteed to be a valid confidence interval for $\theta$.
[The O.P.'s interval is valid in the technical sense that matters here.]
One way to fix this issue would be to derive a different confidence
interval that is valid for all values of $Z$ and $Y$, rather than just
for those satisfying $Z - Y \le a\sqrt[n]\gamma$.
[Again, the O.P.'s interval is valid for all $Z$ & $Y$, & $Z - Y \le a\sqrt[n]\gamma$ is the problematic case.]
To do this, you
could start by deriving the distribution of $Z$ and $Y$ directly,
rather than using the fact that they are sufficient statistics for
$\theta$. This would allow you to find a valid confidence interval for
$\theta$ without making any assumptions about the relationship between
$Z$ and $Y$.
[The distribution of $Z$ & $Y$ is what it is, regardless of how you derive it; their being sufficient has nothing to do with it (nor did the O.P. claim otherwise).]
Alternatively, you could try to find a different set of sufficient
statistics for $\theta$ that do not have the issue you have
identified. This would also allow you to derive a valid confidence
interval for $\theta$.
[Finding different sufficient statistics loosely describes what you need to do, but too loosely to be of any help.]
I've carefully read a score of answers now; a pool of those generously provided by new users over the last week or so, and others generated by me from actual C.V. questions. One could be called useful†; the rest either missed the point, begged the question, or else were riddled with contradictions, non-sequiturs, or brazen confabulations. ChatGPT seems to do anything like a decent job only of answering straightforward, generic questions—of the kind we'd tend to say showed a lack of research in any case.
† The 'useful' one did give a correct explanation of when a paired vs an unpaired test would be appropriate in the particular context; but didn't pick up, as a human would have done, that the destructive testing process described ruled out the obvious way of carrying out a paired t-test, or suggest ways of pairing different experimental units.
P.S. A year & a half later, Google's Gemini is no more use when asked the same question:
The issue arises because the confidence interval you constructed, [Z -
ad_2, Y - ad_1], relies on the assumption that Z (the maximum) will
always be greater than Y (the minimum) within the sample.
Funnier perhaps.
