<p>I'm sharing an example—of an answer to <a href="https://stats.stackexchange.com/q/66407/17230">Confidence interval for Uniform($\theta$, $\theta + a$)</a> generated by ChatGPT. I think it will be agreed that it's either wrong or nonsensical in large part &amp; entirely unhelpful; while being to all appearances worth spending one's time in reading.</p>
<blockquote>
<p>The issue here is that the confidence interval you derived relies on
the assumption that <span class="math-container">$Z - Y \le a\sqrt[n]\gamma$</span>.</p>
</blockquote>
<p>[It's not an assumption at all, but a statement of when the confidence set is empty.]</p>
<blockquote>
<p>If this assumption
does not hold, then the interval <span class="math-container">$[Z -ad_2, Y -ad_1]$</span> is not
guaranteed to be a valid confidence interval for <span class="math-container">$\theta$</span>.</p>
</blockquote>
<p>[The O.P.'s interval <em>is</em> valid in the technical sense that matters here.]</p>
<blockquote>
<p>One way to fix this issue would be to derive a different confidence
interval that is valid for all values of <span class="math-container">$Z$</span> and <span class="math-container">$Y$</span>, rather than just
for those satisfying <span class="math-container">$Z - Y \le a\sqrt[n]\gamma$</span>.</p>
</blockquote>
<p>[Again, the O.P.'s interval is valid for all <span class="math-container">$Z$</span> &amp; <span class="math-container">$Y$</span>, &amp; <span class="math-container">$Z - Y \le a\sqrt[n]\gamma$</span> is the problematic case.]</p>
<blockquote>
<p>To do this, you
could start by deriving the distribution of <span class="math-container">$Z$</span> and <span class="math-container">$Y$</span> directly,
rather than using the fact that they are sufficient statistics for
<span class="math-container">$\theta$</span>. This would allow you to find a valid confidence interval for
<span class="math-container">$\theta$</span> without making any assumptions about the relationship between
<span class="math-container">$Z$</span> and <span class="math-container">$Y$</span>.</p>
</blockquote>
<p>[The distribution of <span class="math-container">$Z$</span> &amp; <span class="math-container">$Y$</span> 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).]</p>
<blockquote>
<p>Alternatively, you could try to find a different set of sufficient
statistics for <span class="math-container">$\theta$</span> that do not have the issue you have
identified. This would also allow you to derive a valid confidence
interval for <span class="math-container">$\theta$</span>.</p>
</blockquote>
<p>[Finding different sufficient statistics loosely describes what you need to do, but too loosely to be of any help.]</p>
<p>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<sup>†</sup>; 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.</p>
<hr />
<p>† 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>
