What is a 99.9 confidence interval?

As a subject matter expert in statistics, I often encounter questions about confidence intervals and their significance in data analysis. A 99.9% confidence interval is a statistical concept that reflects the range where we expect the true population parameter to lie with a high degree of certainty. It is used to estimate an unknown parameter based on sample data. The concept of a confidence interval is closely tied to the idea of a hypothesis test. When we conduct a statistical test, we are often interested in determining whether there is enough evidence to support a certain claim about a population. A confidence interval provides a range of values that we believe includes the true value of the parameter, with a certain level of confidence. ### How to Calculate a Confidence Interval To calculate a confidence interval, you need to follow these steps: 1. Determine the Confidence Level: The confidence level is the probability that the confidence interval will contain the true population parameter. A 99.9% confidence level means that if we were to take an infinite number of samples and calculate a confidence interval for each, 99.9% of those intervals would contain the true population parameter. 2. Select the Appropriate Statistic: For a mean, you would use the sample mean. For a proportion, you would use the sample proportion. 3. Calculate the Margin of Error: The margin of error is calculated using the standard error of the statistic and a critical value from the appropriate distribution (often the normal or t-distribution). For a 99.9% confidence interval, the critical value will be larger than for lower confidence levels, reflecting the greater precision required. 4. Compute the Interval: The confidence interval is computed as the sample statistic plus or minus the margin of error. ### The Role of Sample Size The sample size plays a crucial role in the width of the confidence interval. For small samples, the critical value from the t-distribution is higher than the corresponding z-score from the standard normal distribution. This means that for small samples, the confidence interval will be wider than for larger samples at the same confidence level. This is because smaller samples are less representative of the population, leading to greater uncertainty. ### Example Let's consider an example to illustrate this. Suppose we want to estimate the average height of a certain population. We take a small sample of 10 individuals and find the sample mean height to be 170 cm with a standard deviation of 10 cm. 1. For a 99.9% confidence interval, we would look up the critical value for a t-distribution with 9 degrees of freedom (sample size - 1). This value is higher than the z-score that would be used for a larger sample size. 2. The standard error (SE) is calculated as the standard deviation divided by the square root of the sample size: SE = 10 / √10 ≈ 3.16. 3. The margin of error (ME) is calculated by multiplying the standard error by the critical t-value: ME = SE * t-value. 4. The confidence interval is then calculated as the sample mean ± margin of error: (170 - ME, 170 + ME). ### Conclusion A 99.9% confidence interval is a powerful tool in statistics that provides a high level of certainty about the parameter estimate. It is important to understand that a higher confidence level does not necessarily mean better; it often comes at the cost of a wider interval, which can be less informative. The choice of confidence level should be based on the specific requirements of the analysis and the trade-offs between precision and certainty.

For small samples the t value is higher than the Z value what logically means that the confidence interval for smaller samples with the same confidence level is larger. Z values for matching 90%, 95%, 99% and 99.9% confidence levels are listed in the Table 1.