Does a bigger confidence level have a smaller or bigger confidence interval?

Smaller!  I GOT YOU! I think you mean "there is a smaller chance of obtaining an observation outside that interval". Unfortunately, a Confidence Interval may not mean what it appears to mean, due to technical, statistical issues, but in general the narrower the interval (at a given confidence level) the less uncertainty there is about the results. There are many threads on this site discussing what a Confidence Interval means (as opposed to, say, a Credible Interval). We're not even getting into Predictive Intervals

Step-by-step explanation: i hope this helps let me know if it dues! :)