What does it mean to have a narrow confidence interval?

As a domain expert in statistics, I often encounter the concept of confidence intervals in the context of inferential statistics. A narrow confidence interval is a significant topic within this field, and it has profound implications for the reliability and precision of statistical inferences. When we talk about a confidence interval, we're referring to a range of values that, based on a statistical model, is likely to contain an unknown population parameter. The 95% confidence interval is a common choice because it provides a balance between precision and the breadth of the interval. It means that if we were to take many samples and construct a confidence interval for each, approximately 95% of those intervals would contain the true population mean. The width of the confidence interval is influenced by several factors, including the sample size, the variability within the data, and the level of confidence chosen. When we say that a confidence interval is narrow, we mean that the range of values is relatively small. This typically indicates a higher degree of precision in the estimate of the population parameter. Here's why a large sample size often leads to a narrow confidence interval: 1. Law of Large Numbers: As the sample size increases, the sample mean tends to get closer to the population mean due to the law of large numbers. This reduces the variability in the estimate. 2. Standard Error: The standard error, which is the standard deviation of the sampling distribution of the mean, decreases with an increase in sample size. A smaller standard error directly translates to a narrower confidence interval. 3. Reduced Sampling Fluctuation: Larger samples are less susceptible to sampling fluctuations, which can cause wide swings in the estimate of the mean with smaller samples. 4. Statistical Power: With a larger sample, the statistical power to detect an effect, if one exists, is increased. This means that the study is more likely to correctly reject a false null hypothesis, which can also contribute to a narrower confidence interval if the true effect is being estimated. However, it's important to note that a narrow confidence interval does not necessarily mean that the interval is the "correct" one or that the true parameter lies exactly at the center of the interval. It simply means that we have a more precise estimate based on the data we have. The interval is an estimate, and there is always a chance, albeit small in the case of a 95% confidence interval, that the true parameter does not lie within the interval. In conclusion, a narrow confidence interval is a desirable outcome in statistical analysis as it reflects a more precise estimate of the population parameter. It is achieved through careful study design, a large sample size, and an understanding of the underlying statistical principles.

A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. ... With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.