Interpreting a confidence interval involves understanding the range of values and the confidence level. For example, if a 90% confidence interval for the population mean travel time is [54.085, 65.915], it means we are 90% confident that the true population mean lies within this range. The confidence level (e.g., 90%) indicates the probability that the interval contains the true mean if the sampling process is repeated multiple times. It does not guarantee that the true mean is within the interval for a specific sample. A narrower interval suggests higher precision, while a wider interval reflects greater uncertainty. Factors like sample size, variability, and confidence level affect the width of the interval. Always ensure the sample is random and meets the assumptions for constructing the interval.

A confidence interval for a population mean provides a range of values within which the true population mean is likely to fall, given a specified confidence level (e.g., 90%, 95%). To construct it, use the sample mean () as a point estimate and calculate the margin of error () using the formula:

Here, is the critical z-value, is the population standard deviation, and is the sample size. If is unknown, use the sample standard deviation () and the t-distribution instead. The confidence interval is then calculated as:

This provides the lower and upper bounds of the interval.

The choice between the z-distribution and t-distribution depends on whether the population standard deviation () is known. If is known, use the z-distribution, as it assumes a normal distribution of the population. The margin of error is calculated using:

If is unknown, use the t-distribution, which accounts for additional variability due to estimating with the sample standard deviation (). The margin of error is calculated using:

The t-distribution is used when the sample size is small () or is unknown.