A 95% confidence interval means that if you repeat exactly the same process/experiment many times, 95% of the time the population parameter, e.g. the mean, will fall within that interval. On a practical level, we can say that we are 95% confident the population parameter is within that interval.
One metaphor I like to use to help understand why a confidence interval gets wider as we increase the level of confidence is that of a basketball hoop. The standard hoop is 18 inches in diameter. If a player hits 90% of her free throws using a standard 18” rim, wouldn’t you think her success rate would increase as the size of the hoop increased beyond 18” and her success rate would decrease as the hoop got smaller than 18”? With a 20-inch diameter hoop, her success rate might increase to 95%. With a 24-inch diameter, her free throw success rate might be 99%. If the hoop were only 15 inches, her success rate might fall to 80%.
So it is with confidence intervals. The more confident we want to be, the wider the interval must be.
This is true because of the formula for the margin of error, E = zc*σ/sqrt(n). Assuming the standard deviation σ and sample size n stay constant, E increases as the critical value of z increases. Z-critical increases as we increase the desired confidence level, c.
Here is how to find the critical value of z using StatCrunch. Granted, you will soon memorize the critical values for standard confidence levels of 90%, 95% and 99%, but you may be asked to find different values on quizzes and exams, e.g. 98% or 88%.
Use the steps for using the normal calculator to find z-scores but select the “Between” side (step 4). Enter the confidence level, c, in decimal format, e.g. here c= 95% is 0.95, in the window at step 5, and click compute. The red area is the 95% and the remaining 5% is split between the two tails. You will have the critical values of z, here +1.96 and – 1.96.
On the left, is the normal calculator set up to find the critical values for a confidence level of 88%. You should be able to see by comparing the two graphs that the 95% interval is wider than the 88% interval because the z-critical is larger for the 95%.
Hope this helps.