I have two questions about confidence intervals:
Apparently a narrow confidence interval implies that there is a smaller chance of obtaining an observation within that interval, therefore, our accuracy is higher.
Also a 95% confidence interval is narrower than a 99% confidence interval which is wider.
The 99% confidence interval is more accurate than the 95%.
Can someone give a simple explanation that could help me understand this difference between accuracy and narrowness?
The 95% is not numerically attached at all to how confident you are that you’ve covered the true effect in your experiment. Perhaps recognizing that “interval using 95% coverage range calculation” might be a more accurate name for it. You can make the choice to decide that the interval contains the true value; and you’ll be right if you do that consistently 95% of the time. But you really don’t know how likely it is for your particular experiment without more information.
Your first query conflates two things and misuses a term. No wonder you’re confused. A narrower confidence interval may be more precise but, when calculated the same way, such as the 95% method, they all have the same accuracy. They capture the true value the same proportion of the time.
Also, just because it’s narrow doesn’t mean you’re less likely to encounter a sample that falls within that narrow confidence interval. A narrow confidence interval can be achieved one of three ways. The experimental method or nature of the data could just have very low variance. The confidence interval around the boiling point of tap water at sea level is pretty small, regardless of the sample size. The confidence interval around the average weight of people might be rather large because people are very variable but one can make that confidence interval smaller by just acquiring more observations. In that case, as you gain more certainty about where you believe the true value is, by collecting more samples and making a narrower confidence interval, then the probability of encountering an individual in that confidence interval does go down. (it goes down in any case when you increase sample size, but you may not bother collecting the big sample in the boiling water case). Finally, it could be narrow because your sample is unrepresentative. In that case you are actually more likely to have one of the 5% of intervals that does not contain the true value. It’s a bit of a paradox regarding CI width and something you should check by knowing the literature and how variable this data typically is.
Further consider that the confidence interval is about trying to estimate the true mean value of the population. If you knew that spot on then you’d be even more precise (and accurate) and not even have a range of estimates. But your probability of encountering an observation with that exact same value would be far lower than finding one within any particular sample based CI.
Q2: A 99% confidence interval is wider than a 95%. Therefore, it’s more likely that it will contain the true value. See the distinction above between precise and accurate, you’re conflating the two. If I make a confidence interval narrower with lower variability and higher sample size it becomes more precise, the likely values cover a smaller range. If I increase the coverage by using a 99% calculation it becomes more accurate, the true value is more likely to be within the range.