Suppose I am a consultant and I want to explain to my client the usefulness of confidence interval. The client says to me that my intervals are too wide to be useful and he would prefer to use ones half as wide.
How should I respond?
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It depends on what the client means by "useful". Your client's suggestion that you arbitrarily narrow the intervals seems to reflect a misunderstanding that, by narrowing the intervals you've somehow magically decreased the margin of error. Assuming the data set has already been collected and is fixed (if this isn't the case, @shabbychef's joke in the comments gives you your answer), any response to your client should emphasize and describe why there's no "free lunch" and that you are sacrificing something by narrowing the intervals.
Specifically, since the data set is fixed, the only way you can reduce the width of the confidence interval is by decreasing the confidence level. Therefore, you have the choice between a wider interval that you're more confident contains the true parameter value or a narrower interval that you're less confident about. That is, wider confidence intervals are more conservative. Of course, you can never just optimize either width or confidence level mindlessly, since you can vacuously generate a $100 \%$ confidence interval by letting it span the entire parameter space and can get an infinitely narrow confidence interval, although it will have $0 \%$ coverage.
Whether or not a less conservative interval is more useful clearly depends both on the context and how the width of the interval varies as a function of the confidence level, but I'm having trouble envisioning an application where using a much lower confidence level to obtain narrower intervals would be preferable. Also, it's worth pointing out that the $95 \%$ confidence interval has become so ubiquitous that it will be hard to justify why you're, for example, using a $60\%$ confidence interval.
I would suggest it entirely depends on what your client wants to use the confidence intervals for.
I would be quite proud of myself if I managed to keep "So run a bigger study" from being the first thing out of my mouth.
You provide a confidence interval at a certain standard level such as 90% or 95%. The client can judge whether or not the interval is too wide to be useful. But of course that does not mean that you can shorten it to make it useful. You can suggest that increasing the sample size will decrease the width of an interval at a given confidence level as it decreases roughly by a factor of the square root of the sample size.