When would I want a small confidence interval and when would I be satisfied with a larger one? When is a larger sample important?
Let's see if we can shed a little light on your two situations. As an applied statistician in academe and business I am often asked the same questions you have presented here. First, however, follow along with me so that I can give you a little reasoning behind the need for confidence intervals and sample size.
In all statistical processes, parametric or non-parametric, accuracy is what one strives to obtain. Measurement numeric values by themselves carry no real meaning. It is not until you arrange these numbers in some sort of fashion and insert them into a certain particular statistical process that something of value happens. Now...follow closely here...once you place your measurement numeric values in a formula to test for differences, relationships and or effects the resulting numeric value is not longer a measurement value but a mathematical values that represents the strength of the relationship, difference, or effect that you set up to investigate. Again...follow along with me now. :-)
The mathematical value by itself is meaningless until we determine it's strength. And, we do this by establishing confidence levels to see if the mathematical value received from the statistical data process formula is, indeed, a representation of a statistically significant difference, relationship or effect that we tested for and just how much of the difference, relationship or effect is due to chance or random error. Okay so far? :-) Good...let's go on then.
As a research investigator you must always set the confidence level for the research study ...
This solution provides an explanation as to why someone would prefer a small confidence interval over a large one.